Galatius (in his talk) has made very interesting remarks about the stable range of some classifying spaces of groups. To be more concrete, I will mention two examples to illustrate his (?) point of view.
Example 1
Let $Br_{n}$ the $n$-braided groups then the integral homology $H_{\ast}(BBr_{n},\mathbf{Z})$ is isomorphic (in stable range) to $H_{\ast}(\Omega^{2}_{0}S^{2},\mathbf{Z})$ where the $\Omega^{2}_{0}S^{2}$ is the connected component of double loop space of the sphere. $$ BBr_{\infty}\simeq_{H_{\ast}} \Omega^{2}_{0}S^{2} $$
Example 2
In the case of the group of automorphisms of the free group with $n$-generators $Aut(F_{n})$ he proved that there is a homology isomorphism (in stable range) between $H_{\ast}(BAut(F_{n}),\mathbf{Z})$ and $H_{\ast}(\Omega^{\infty}_{0}S^{\infty},\mathbf{Z})$ i.e., $$B[colim_{n} Aut(F_{n})]\simeq_{H_{\ast}}\Omega^{\infty}_{0}S^{\infty} $$
Question: Is there examples of a sequence of (topological) groups $\dots\rightarrow G_{n}\rightarrow G_{n+1}\dots$ such that we have a homology isomorphism between $$BG_{\infty}\simeq_{H_{\ast}}\Omega^{i}_{0}X$$ where $X$ is not a loop space and $2<i<\infty$. In particular is there a sequence of (topological) groups $\dots\rightarrow G_{n}\rightarrow G_{n+1}\dots$ such $$BG_{\infty}\simeq_{H_{\ast}}\Omega^{n}_{0}S^{n}$$
Examples with a geometric meaning will be appreciated. If I wrote something wrong, do please correct me.
